On Σ-products of Σ-spaces

On the dimension of limits of inverse systems

“…REMARK 2. For a Cartesian product X as in Proposition 3, it follows from [PP,Proposition 2], [Tk,Theorem 1] and [Y,Theorem 1] that each open FCT-set in X is a cozero-set. So, Proposition 3 is a generalization of [KI,Theorem 1].…”

Countable paracompactness of Σ-products

“…However, the countable tightness is no longer a necessary condition for a Z-product of paracompact Z-spaces to be normal, because there exists a collectionwise normal Z-product of Mx -spaces which has no countable tightness [18]. Moreover, since there exists a nonnormal Z-product of Mxspaces [18], in Question 1 the assumption of countable tightness cannot be dropped. On the other hand, Rudin [16] proved that any Z-product of metric spaces is shrinking and hence countably paracompact.…”

“…Our proof of Theorem 1 is based on the idea in Yajima [18,20] and we shall use the following fact: a space X is normal if and only if for every pair A, B of disjoint closed subsets of X there exists a er-locally finite open cover ^ of X such that either Ur\A = 0 or UnB = 0 for every U £Î7.…”

“…Question 1 has been answered positively. In fact Yajima [18] even proved (iii) A Z-product of paracompact Z-spaces is (collectionwise) normal if it has countable tightness.…”

“…In connection with the above results, the following Questions 1 and 2 are considered by Yajima [18] and Kodama, respectively. Question 1.…”